Ideal Quantum Gases: The General Framework

The distributions of the previous lessons give the mean occupation of one mode. Turning them into the thermodynamics of a gas means summing over all modes, and for a macroscopic box the levels are so dense that the sum becomes an integral weighted by a density of states. The integrals that result are the same two special functions for every ideal quantum gas, the Bose function $g_{\nu }(z)$ and the Fermi function $f_{\nu }(z)$, and the relation between pressure and energy follows from the shape of the density of states alone, not from which statistics governs. This lesson assembles that shared machinery. The four applications that follow it (blackbody radiation, phonons, Bose-Einstein condensation, and the degenerate Fermi gas) are each obtained by choosing the density of states and the chemical potential and reading off the same formulas.

From the mode sum to an energy integral

The extensive quantities of the ideal quantum gas are sums over single-particle levels weighted by the mean occupation,

For free particles in a cube of side $L$ with periodic boundary conditions the allowed wavevectors are spaced by $2\pi /L$, so a volume $(2\pi /L{)}^3$ of $\vec{k}$-space holds one state. The level spacing shrinks as $L\to \infty$, and the sum passes to an integral over $\vec{k}$, then over energy once the states are grouped by $\epsilon$:

with $g_s=2s+1$ the spin multiplicity. For nonrelativistic particles, $\epsilon ={\hslash }^2{k}^2/2m$, and counting the $\vec{k}$-shell of energy $\epsilon$ gives the density of states

The ${\epsilon }^{1/2}$ growth is the geometric fact that higher energy shells in momentum space have more states; it is the same factor that shaped the Maxwell speed distribution. Replacing the discrete spectrum by this smooth weight is exact in the thermodynamic limit and drops only the single lowest level — an omission that is harmless for fermions but is precisely what must be restored by hand for a condensing Bose gas.¹

The Bose and Fermi functions

Substituting the density of states and the mean occupation, and changing variables to $x=\beta \epsilon$ with fugacity $z=e^{\beta \mu }$, expresses the number density and pressure through two standard functions. Define

the Bose-Einstein and Fermi-Dirac functions, convergent for $0\le z\le 1$ and $z\ge 0$ respectively. Both start as $z$ at small fugacity, the Maxwell-Boltzmann limit. With these, the ideal quantum gas is summarized in a few lines:

where $h_{\nu }$ stands for $g_{\nu }$ (bosons) or $f_{\nu }$ (fermions) and $\lambda$ is the thermal wavelength. The index shifts by one between the number and the pressure because the pressure integral carries one extra power of energy from the integration by parts below. At $z=1$ the Bose function reaches the finite value $g_{3/2}(1)=\zeta (3/2)=2.612$,² and that this maximum is finite is the mathematical origin of Bose-Einstein condensation.³

Pressure and energy from the density of states

The relation between pressure and energy is fixed by the exponent of the density of states, independent of statistics and of temperature. Write $g(\epsilon )=c{\epsilon }^s$; for the nonrelativistic gas $s=\frac{1}{2}$. The grand potential is $-PV=\mp k_{B}T\int g(\epsilon )\ln (1\mp z{e}^{-\beta \epsilon })\mathbf{d}\epsilon$, and integrating by parts moves the log onto the occupation.

The result is stronger than the ideal-gas law it generalizes. It holds for bosons and fermions alike, at any degeneracy, and reduces to $PV=N{k}_{B}T$ only in the classical limit where $U=\frac{3}{2}N{k}_{B}T$. The degeneracy pressure of a cold Fermi gas and the radiation pressure of a photon gas are both instances of it, differing only through $s$.

The four applications as one framework

Everything downstream is a choice of two ingredients: the density of states $g(\epsilon )$, fixed by the dispersion and dimensionality, and the chemical potential $\mu$, fixed by whether the particle number is conserved.

• Photon gas. Photons are not conserved, so $\mu =0$ and $z=1$; the dispersion is $\epsilon =pc$, giving $s=2$ and $PV=\frac{1}{3}U$. The occupation is the pure Bose function at zero chemical potential, and the energy integral is the Planck spectrum.
• Phonons. Lattice vibrations are bosons with $\mu =0$ and a linear acoustic dispersion cut off at the Debye frequency by the finite mode count $3N$. The same $PV=\frac{1}{3}U$ machinery gives the Debye heat capacity.
• Bose gas. Conserved bosons keep $\mu <0$ and $z<1$ until $g_{3/2}(z)$ hits its ceiling $\zeta (3/2)$, at which point the excited states saturate and a condensate forms in the omitted ground level.
• Fermi gas. Conserved fermions have $\mu \to {\epsilon }_F>0$ as $T\to 0$; the Fermi function replaces the Bose function, the occupation approaches a step, and the pressure is the degeneracy pressure.

The photon gas and Bose-Einstein condensation carry the boson branch, and the degenerate Fermi gas carries the fermion branch; each specializes the formulas assembled here.

Summary

• The mode sum becomes ${\int }_0^{\infty }g(\epsilon )\mathbf{d}\epsilon$ with density of states $g(\epsilon )=g_{s}2\pi V(2m{)}^{3/2}{h}^{-3}{\epsilon }^{1/2}$ for a nonrelativistic gas in three dimensions.
• Number, pressure, and energy reduce to the Bose functions $g_{\nu }(z)$ or Fermi functions $f_{\nu }(z)$: $N/V=(g_s/{\lambda }^3)h_{3/2}(z)$ and $P/k_{B}T=(g_s/{\lambda }^3)h_{5/2}(z)$, with $U=\frac{3}{2}PV$ nonrelativistically.
• Integration by parts gives $PV=U/(s+1)$ for $g\propto {\epsilon }^s$: $\frac{2}{3}U$ nonrelativistic, $\frac{1}{3}U$ ultrarelativistic, independent of statistics.
• The finite ceiling $g_{3/2}(1)=\zeta (3/2)=2.612$ foreshadows condensation.
• Choosing $g(\epsilon )$ and $\mu$ specializes the framework to the photon gas, phonons, the Bose gas, and the Fermi gas.